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Now j k extends f , which is a contradiction. For the case G = B, the function j G extends f and gives the contradiction. 7. MODEL COMPLETIONS 57 The following lemma completes the proofs that each of the theories DLO, ACF and RCF admit elimination of quanti ers. Lemma 19. Each of the following three pairs of theories T and T satisfy condition (3) of Blum's Test. (1) T = LOR theory of linear orderings. T = DLO, theory of dense linear orderings without endpoints. (2) T = FEI, theory of elds. T = ACF, theory of algebraically closed elds.

Each model of T is existentially closed. (3) for each formula '(v0 : : : vp ) of L there is a universal formula (v0 : : : vp ) such that T j= (8v0 : : : 8vp )(' $ ) (4) for all models A and B of T , A B implies A B. Proof. (1) ) (2): Let A j= T and B j= T with A B. Clearly AA j= 4A it is also easy to see that BA j= 4A. Now by (1), T 4A is complete and both AA and BA are models of this theory so they are elementarily equivalent. 32 4. MODEL COMPLETENESS 33 So let be any sentence of LA (existential or otherwise).

We use conditions (1) and (2) to prove the following: Claim. T has existentially closed models of each in nite size . Proof of Claim. By the Lowenheim-Skolem Theorems we get A0 j= T with jA0 j = . We recursively construct a chain of models of T of size A0 A1 : : : An An+1 with the property that if B j= T and An+1 B and is an existential sentence of Th(BAn ), then (An+1 )An j= . Suppose An is already constructed we will construct An+1. Let n be a maximally large set of existential sentences of LAn such that for each nite 0 n there is a model C for LAn such that C j= 0 T 4An By compactness T n 4An has a model D and without loss of generosity An D.